LMFDB - L-function 2-48e2-24.11-c3-0-89

Dirichlet series

L(s) = 1

− 17.6·5^-s − 14.9i·7^-s − 55.1i·11^-s + 9.02i·13^-s + 14.1i·17^-s − 139.·19^-s − 86.1·23^-s + 186.·25^-s + 3.64·29^-s − 273. i·31^-s + 264. i·35^-s − 405. i·37^-s − 344. i·41^-s − 241.·43^-s − 297.·47^-s + ⋯

L(s) = 1

− 1.57·5^-s − 0.808i·7^-s − 1.51i·11^-s + 0.192i·13^-s + 0.202i·17^-s − 1.68·19^-s − 0.781·23^-s + 1.49·25^-s + 0.0233·29^-s − 1.58i·31^-s + 1.27i·35^-s − 1.80i·37^-s − 1.31i·41^-s − 0.857·43^-s − 0.924·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.169 - 0.985i)\, \overline{\Lambda}(4-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.169 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$2304$ = $2^{8} \cdot 3^{2}$
Sign:	$-0.169 - 0.985i$
Analytic conductor:	$135.940$
Root analytic conductor:	$11.6593$
Motivic weight:	$3$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{2304} (1151, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 2304,\ (\ :3/2),\ -0.169 - 0.985i)$

Particular Values

$L(2)$	$\approx$	$0.3777407773$
$L(\frac12)$	$\approx$	$0.3777407773$
$L(\frac{5}{2})$		not available
$L(1)$		not available

Euler product

$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} $

	$p$	$F_p(T)$
bad	2	$ 1 $
	3	$ 1 $
good	5	$ 1 + 17.6T + 125T^{2} $
	7	$ 1 + 14.9iT - 343T^{2} $
	11	$ 1 + 55.1iT - 1.33e3T^{2} $
	13	$ 1 - 9.02iT - 2.19e3T^{2} $
	17	$ 1 - 14.1iT - 4.91e3T^{2} $
	19	$ 1 + 139.T + 6.85e3T^{2} $
	23	$ 1 + 86.1T + 1.21e4T^{2} $
	29	$ 1 - 3.64T + 2.43e4T^{2} $
	31	$ 1 + 273. iT - 2.97e4T^{2} $
	37	$ 1 + 405. iT - 5.06e4T^{2} $
	41	$ 1 + 344. iT - 6.89e4T^{2} $
	43	$ 1 + 241.T + 7.95e4T^{2} $
	47	$ 1 + 297.T + 1.03e5T^{2} $
	53	$ 1 + 602.T + 1.48e5T^{2} $
	59	$ 1 + 630. iT - 2.05e5T^{2} $
	61	$ 1 + 237. iT - 2.26e5T^{2} $
	67	$ 1 - 397.T + 3.00e5T^{2} $
	71	$ 1 + 778.T + 3.57e5T^{2} $
	73	$ 1 - 1.15e3T + 3.89e5T^{2} $
	79	$ 1 + 127. iT - 4.93e5T^{2} $
	83	$ 1 - 301. iT - 5.71e5T^{2} $
	89	$ 1 - 338. iT - 7.04e5T^{2} $
	97	$ 1 - 1.61e3T + 9.12e5T^{2} $
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$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$

Imaginary part of the first few zeros on the critical line

−8.158818322415748547033669801279, −7.48419452762562815150776688887, −6.60216821830797074031812359082, −5.86235804005178156612251605729, −4.61260682454703644349876520004, −3.85017638944115961429463002746, −3.52528304077593245295635661061, −2.09687976835111079018725174035, −0.49461071374444284453177801677, −0.14869541838229273245427498992, 1.51880835321507029127735341213, 2.61538005269625185867634134052, 3.57109100180965436476023471308, 4.56413536823204026314839116015, 4.89412223388325789919396585391, 6.33118498929834446103156663002, 6.89184308540478518565197151658, 7.85791582351238071064829077207, 8.283281647732335165477414043957, 9.029397119306985323052886757435

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

\(L(2)\)	\(\approx\)	\(0.3777407773\)
\(L(\frac12)\)	\(\approx\)	\(0.3777407773\)
\(L(\frac{5}{2})\)		not available
\(L(1)\)		not available

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